digital calculation of earthing systems in nonuniform soil

Archly fiir Elektrotechnik 62 (1980) 19-- 24 ARCHIV FOR ELEKTROTECHNIK �9 by Springer-Verlag 1980

Digital Calculation of Earthing Systems in Nonuniform Soil

J . M . N a h m a n

Contents: The mathematical model of earthing systems consisting of straight-line elements ill the nonuniform soil is developed. The method presented enables the evaluation of all earthing system characteristics by applying a general algorithm, appropriate for digital computing, based upon elementary data of the earthing system and soil configuration.

Digitale Berechnung yon Erdungsanlagen im inhomo- genen Erdreich

~)bersieht: Es wird ein mathematisches Modell ffir die aus geradlinigen Elementen zusammengesetzten Erdungsanlagen in gesehichtetem Erdreich entwickelt. Die vorgeschlagene Methode erm6glicht die Berechnung aller Kennlinien der I;rdungsanlage bei Anwendung eines allgemeinen Verfahrens, das auf der elementaren Daten fiber die Konfiguration der Erdungsanlage und des Erdreiches beruht.

r

List of Symbols

u electric potential (voltage) e c contact voltage ep pace voltage i, i~ current R resistance to earth r self or mutual resistance ~1 earth resistivity of the upper soil layer ~2 earth resistivity of the bottom soil layer h depth of the upper soil layer l length of the earthing system element d diameter of the earthing system element V general symbol for the rectangular space coor-

dinates X, Y, Z of a point R(M, K) distance between points M and K Other symbols are defined in the text.

1 I n t r o d u c t i o n

Var ious me thods have been p r o p o s e d for the ana lys i s of e a r t h i n g sys tems in nonun i fo rm soil in the recen t years .

The m a t h e m a t i c a l mode l has been deve loped for ca lcu la t ing the res i s tance to e a r t h of concent r ic

r ings in the two- l aye r soil [1], g iv ing a fa i r a p p r o x i - m a t i o n of squa re - shaped conve t iona l gr ids for such purposes . However , the mode l i n v e s t i g a t e d canno t be expec t ed to p rov ide a s a t i s f a c to ry a p p r o x i m a t i o n of the e a r t h surface p o t e n t i a l d i s t r i bu t i on which is of in te res t for the e s t ima t ion of the pace- and c o n t a c t - vol tages .

Ve ry in t e r e s t i ng m e t h o d s for e v a l u a t i n g t i le r e s i s t i v i t y tes t s for des igning s t a t i on e a r t h i n g sys tems in the nonun i fo rm soil have been also p r o p o s e d in [2, 3]. The a p p r o x i m a t i o n s a d o p t e d in the app ro - aches m e n t i o n e d l imi t the a rea of the i r app l i ca t i on [4].

The comple te , t he o re t i c a l l y s t r o n g l y founded mode l of e a r t h i n g sys tems in the t w o - l a y e r soil, deve loped b y A. I. J a k o b s and his co -au thors [4], enables the s tudies of e a r t h i n g sys tems cons is t ing of pa ra l l e l and pe rpe nd i c u l a r s t r a igh t - l ine e lements . The se l f -and mu tua l - r e s i s t ances of the e a r t h i n g sy s t em e lements are d e t e r m i n e d b y means of com- pos i te ana ly t i c fo rmulas deve loped a p p l y i n g a modi - f ica t ion of the m e a n - p o t e n t i a l me thod .

This p a p e r p resen ts a m e t h o d based on the numer i ca l eva lua t i on of the res is tances of e a r t h i n g sy s t em elements , enab l ing the ana lys i s of ea r th ing sys tems wi th s t r a igh t - l ine e lements in an a r b i t r a r y conf igura t ion .

2 M a t h e m a t i c a l M o d e l of E a r t h i n g S y s t e m

Let us cons ider an e a r t h i n g sy s t em cons is t ing of n s t r a i g h t l ine e lements .

The p o t e n t i a l u of the ea r th ing sy s t em and the v e c t o r i of the cu r ren t s f lowing in to the e a r t h f rom the e lements are connec ted b y the r e l a t i o n

Eu := r i . (t)

20 Archly ffir Elektrotechnik 62 (1980)

The vector E has n unit components. The elements r~ of the symmetric nonsingular matr ix r are self (j = k) and mutual (j 4= k) resistances of earthing system elements. The resistances mentioned above are the total values for a two-layer soil, involving the influence of the earthing system images.

The ground-fault current i~ which is to be di- scharged from the earthing system into the earth and the currents i are connected by the expression

ig ~ ~ i k , (2) k = l

i k being the current of the element k. Dividing both sides of Eq. (t) by u we obtain a

system of lineaI equations for i J u , k = 1 . . . . , n .

Since rkk ~ rkj, k, j = 1 . . . . . n, this system of equations can be easily solved applying the well known GauB -- Seidel method I5].

The resistance to earth of the earthing system will be, with respect to (2),

R - q (3)

and the the potential

u = R i g . (4)

The potential u r in an arbi t rary point T , caused by the earthing system, equals to

UT = ~ rkTi k , (5)

parameter rkT representing the mutual resistance of the earthing system element k and the point T for the two-layer soil.

Eqs. (3) -- (5) determine all necessary earthing system characteristics.

3 M u t u a l - a n d S e l f - r e s i s t a n c e s

Let us subdivide the earthing system into elements, each being with its whole length in one of the soil layers. The element lengths selected depend on the desired accuracy of calculations.

3.1 R e s i s t a n c e s f kT

The electric potential at a point T caused by a straight-line element of the earthing system, burried in the two-layer soil, can be calculated by supper- position of the potentials from the corresponding infinite sequences og images laying in the uniform soil [6], as shown in Fig. t.

h ~ D g2ik

~ . . . . . g Jk h g " " ~

h ~ ~ " P1 gtk

h " B ~ ~

~ g ( l + g ) i k B ~

i i . ~ l ~ l l / / / / , r / / H l , , / / / i , . ~ l I I /H~ / ' ~ l , ' , ' l ' ~ ' / , t l ~ / r / , t IH .

z '~" h P~ z ~ k P~

h ~ " ~ i k h

C _ . _ . ,..... gZi k A ~ h - - __ h _ _ " " g ' ~ " , g } i k

&

h

h ~ ~ " g ( 1+ g } i k PI h

h " ~ ~ ~ ( l + g ) i k h

�9 T ~ , , T

Z Z o d

...... g { 1 -g2} ik

~ ( 1 - g 2 ) i k B ~ "Pz

t / / , , / ~ / ~ / / / / , , , , / , , , , / / / l / / , , / / /

Fig. 1. Image sequences for calculating the potential caused by a straight-line earth electrode in two-layer soil in various cases, - - electrode, - ........ image

Fig. I a displays the image disposition for calculation of the potential in a point T , for the earthing system element k and the point T, both laying in the upper layer. Fig. I b is relevant to the case when the point T is in the upper and the earthing system element in the bot tom soil layers, while Fig. I c to the inverse situation. Fig. t d gives the dispositions of the images for the point T and earthing system element both laying in the bot tom layer. The non- uniformity factor g is defined as :

g = (e. - el) (02 + 01) -1 . (6)

There are generally four different sequences of images, denoted by A, B, C, D.

The Z-coordinates of the end-points of the images for the sequences are given by the relationships

l A -~- Z ~- 2 s h , Z B = - - Z - - 2 s h , (z)

Z c = - - Z + 2 s h , Z D = Z - - 2 s h ,

J. M. Nahman: Digital Calculation of Earthing Systems in Nonuniform Soil 21

where Z denotes the Z-coordinates of the real earthing system element end-points. The integer s represents the general index of the sequences.

The X, Y-coordinates of the image end-points are the same as those for the real element.

From Eq. (7) it follows that the first terms of sequences A and D, corresponding to s = 0, represent the real earthing system element.

The mutual resistance of the element k and a point T for one of the dispositions displayed in Fig. I will be

~kr(s) s = 0

rkr -- ik , (8)

q0~r(S) representing the potential caused by the terms s of the image sequences.

The potential contributed from an image at a point T can be determined applying the general formula for a straight-line conductor in uniform soil

l Oi G + l d z ~/~/In G ~ I +

G - - l ' 5 ' ~o~ = d~ (9)

~ / l n G = < I + 2~'

where

G = R(P, T) + R(Q, T ) , (10)

P and Q being the end-points of the conductor. Symbols i, l, d, ~ denote the conductor 's current, length and diameter and the resistivity of the soil, respectively.

With regard to Eqs. (8) and (9) and Fig. t a, for the mutual earth resistance of the element k and the point T, both laying in the upper layer, we obtain

{&~(o) + G~-(o) + rkT -- 4~lk

+ 2 g'[AkT(S) + Bkv(s) + CkT(S) + D~T(S)] . s - - 1

(11) The parameters AkT(S), BkT(S), C~T(S), D~r(s)

represent the LN-functions in Eq. (9) for the element k images belonging to the sequences A, B, C, D, respectively.

Using the same notat ion as in Eq. (11), for the element k in the bot tom and the point T in the upper layer we can write (Fig. t b)

co

~1 (1 -~ g) Z g~[Akr(a) + BkT(S)] (12) r~r - - 4~lk h = 0

and for the inverse situation (Fig. t c)

oo

~1 (1 -~ g) X gsCBkT(S) ~- DkT(S) "] (13) s=O

When the element k and the point T both lie in the bot tom layer we have, according to Fig. t d,

02 Akr(0) --gC~T(t) + (t g'Bkr(S) ~'kT ~- 4~1 k = �9

(14) In practical calculations infinite series in Eqs.

(t t) to (14) are approximated with a finite number of first terms, guaranteeing the accuracy desired (App. "Evaluat ion of the sums of infinite series").

3.2 Resistances rkj

The mutual earth resistances for two earthing system elements k and j can be evaluated as an arithmetic mean value of the mutual resistances of the element k and several, say, equidistant points laying on the element j . If we take f. e. N 1 such points, the mutual resistance for elements k and j will be

1 N1

rkj = ~ E rk:, (15) 1J=l

where the indices J denote the points of the element j , chosen for calculations.

The coordinates of points d r can be simply determined by means of corresponding end-point coordinates of the element j

V: VpS + (Vos -- Vp:) ] - t (t6)

Symbols Vpj, VOj and Vj denote the coordinates of the element j end-points and point J ,respectively.

The relationships (7) -- (16) show that the evaluation of the resistances rkj can be carried out by applying a general algorithm with end-point coordinates of k and j elements and diameters of k elements as only input data.

Since Eq. (9) involves the case with points J laying on the element k, the same algorithm is valid also for the evaluation of rkk -- the self resistances of earthing system elements.

If we denote the earthing system elements laying in the upper layer by t to m, the resistance matr ix in Eq. 1 will be

r = i . . . . . ( t 7 )

The elements of matrices (a), (b), (c) and (d) are determined by means of the algorithm described priorly, employing Eqs. ( t l) -- (t4), respectively.

The matrices (e) and (b) are connected by the relationship

c : b t (18)

22 Archiv fiir Elektrotechnik 62 (t 980)

which simplifies the calculations. Eq. (18) can be also employed for est imating modelling errors involved by Eq. (t 5). When necessary, the accuracy of the formula ment ioned can be easily improved by increasing N 1.

4 A p p l i c a t i o n E x a m p l e

A grid earthing system combined with vertical electrodes in the two-layer soil, showen in Fig. 2, has been investigated employing the method proposed in ehis paper. Horizontal grid elements are round conductors ~ 7 0 m m 2, the vertical ones, pipes 2 1/2".

The potentia! distribution u T along the line I -- t ' on the earth surface has been calculated and the max imum contact- and pace-voltages determined. The results obtained for various cases are presented in Fig. 3 and Table 1.

z x

E

7

y ~ 0.7m

2ml I t IT2m m

"\\x'~I\ \\\\\\x\ \\\\\\\\\\\\\\ \\\ \ \ x\\\'~ \ \\\ ~2

Fi~. 2. Earthing system sample

{ �9 ~_......2,s

22 3o 35 40 is X-Coordinates d the poinls along fine 1-1'

- I

' 7'6 05 m

Fig. 3. Potential distribution on the earth surface along the line I - - 1" for different cases, as denoted in Table I

One can observe a very unfavourable potential distribution for high ~1/% ratio. I t is interesting to notice also tha t for ~olLo 2 = 30/300 the earthing system analysed has approximate ly the same contact -- voltages as for uniform soil with o = = 30 f~m, despite a cc- -a five times greater resistance to ear th in the first case.

The positive influence of the vertical electrodes can be observed only for high ~1/o2 ratio.

The analysis performed shows also tha t modelling the earthing system with an element per conductor provides a fair approximat ion of the problem for practical investigations.

The condition (t 8) has been satisfactorily fulfilled with N~ = 40.

The simplified flow char t in Fig. 4 describes the calculation procedure. The input end-point coordinates of earthing system elements, Vp, V O, and their diameters are stored as n-component vectors, integers h or j being general indices of the vectors. Inpu t data V r are coordinates of the initial point on l i ne t - - t ' .

T a b l e I

Nr. 01/~ R eJig ep/ig ( r im/am) (a) VIA VIA

Earthing Total system computation configuration time (system

I B M - 370) (min./sec.)

300/30 1.0336 0.79500 0.11991 30/30 0.32038 0.10480 0.02205 30/300 1.6o90 0.11955 0.04845

1' 300/30 t.0155 0.77956 0.12487 2' 30/30 0.31791 0.10125 0.02291 3' 3O/300 1.6059 �9 0.09928 0.05t02

Without vertical t 4/23 electrodes, each conductor 2/35 taken as a. single element I4/59

Without vertical electrodes, 30112 edge-conductors subdivided 3/12 into 3 elements 32/56

4 300/30 0.94278 0.68955 0.10343 5 30/30 0.3t557 0.09972 0.02110 6 30/30O 1.5730 0.10711 0.04576

With vertical electrodes, 25/4t subdivided into 2 elements 3/25 according to soil layers 25/56

J. M. N a h m a n : Digi ta l Calculat ion of lZarthing S y s t e m s in Nonun i fo rm Soil 23

START )

t READ: Vpik jVQ(k),d(kl, V T

'~VT , Vrna• ,O1 , .P2. h

ig , n , rn , N1 , E" I

I CHANGE; k=1~n I

t I CHANGE:j=I--,-n ]

rk] USING rki USING ] rki USING rki USING [11) I as) I I ill13) J{IS}l ] {12)JlS)I (1~ I) ,(15)

t [ COMPUTE ifk)~ R~ U USING {11~ (31, [411 L F

[, CHANGE: k =,--rl J

no

I I

I COMPUTE uT USING (5) l

@ , no

( s,o )

Fig . 4. Simplif ied c o m p u t i n g p rocedure flow cha r t

VT = V T + ~,V T J--

5 C o n c l u s i o n

The method of calculation presented in this paper can be applied to any earthing system configuration consisting of straight-line elements. The algorithm for evaluation of self and mutual resistances of the elements has the same general mathematical form irrespective of the disposition of the elements. The whole calculation of the earthing system can be prepared as a general, easily applicable computer program, with the elementary information on the earthing system and the soil configuration as only input data.

6 A p p e n d i x e s

6. l Potentials Caused from a Straight-line Conductor in Uniform Soll

Let us consider a straight-line cylindrical conductor laying in a homogenous soil with earth resistivity o.

When an uniform density of the current i emer- ging from the conductor axis over its length is

T

F ig . A I , Po t en t i a l d i s t r i bu t ion caused f rom a s t r a igh t - l ine conduc to r in u n i f o r m soil

assumed, the equipotential surfaces will be ellipsoids of revolution about the conductor axis [7]. The end-points P, Q of the axis are the loci of the ellipsoids.

The real conductor is replaced by an ellipsoid of revolution passing through the circumference points T O at the middle of the conductor (Fig. A t).

The equation of the equivalent ellipsoid will be

R(P, T) + R(Q, T) = R(P, To) + R(Q, To). (A t)

We have

R(P,, To) = R(Q, To) : 2 I 07 ~ 2 +-4 l- '

(A 2)

if d ~ . l, which is allways the case. According to (A 2), Eq. (A t) takes the form

d 2 R(P, T) + R(Q, T) = 1 + -27" (A 3)

The potential of a point T in the soil surrounding the conductor equals to

R(P, r) + R(Q, r) + / i .~ in �9 (A 4)

~ T - - 4 . ~ . 1 R ( P , T) q- R(Q, T) - - 1

The point considered does not belong to the equivalent conductor if

d2 R(P, r) + R(Q, T) > l + 27"

All points which for d e

R(P, M) + R(Q, T) <= I + 2--[

belong to the conductor. They all have the same potential which equals to the potential of the points T o. The last one will be, with respect to (A 2) and (A 4),

l + + l

i. e in ~ - - in -

24 Archly far Elektrotechnik 62 (t 980)

6.2 Evaluation of the Sums of Infinite Series

We can write

g~F(s) = ~ g'F(s) + R,~ i = O, t , s = i s ~ i

where co

R,,, = X g 'F(s) . s=m+l

m > i ,

If F(s) is decreasing with s in absolute value and [gl ~ 1, which is the case in our considerations, it will be

IR~r < 0

with oo

b = I F ( m + l ) l ~ I g l ' = IF(m+t)gm+~[ ( t - - lg l ) -1" s=m+l

(a 5) The remainder R,, presents the error if the

infinite series are subst i tuted with their first m terms.

For the relative error

tr, L = IR, l Is : gsFisl + l - -

if Im !

6 < ,=-~fF(s) , (A 6)

we can write

) - 1 ]r,~l C e = 6 Y~ g'F(s) - - ~ (A7)

For a fixed m the paramete r 6 is calculated and the condition (A 6) checked. If satisfied, the relative error is est imated by means of Eq. (A 7).

If

e < q , (A 8)

el ~ 0 denoting the m a x i m u m relative error per- mitted, evaluation of the series is finished. In the case when either tile condition (A 6) or (A 8) is not fulfilled, the number m has to be increased and the procedure repeated.

A c k n o w l e d g m e n t

The author wishes to express his grat i tude to the staff of Electricity board Computer Center, Belgrade, for the help in performing the i l lustrative calculations.

References

t. Zaborsky, J. : Efficiency of grounding grids with nonuniform soil. A I E E Trans. Vol. 74, P. I I I (1955) 1230--1233

2. Thapar, B.; Gross, E. T. t3.: Grounding grids for high voltage stations. IV-Resistance of grounding grids in nonuniform soil. Trans. PAS 68 (t963) 782--788

3. Endrenyi, J . : Evaluat ion of resistivity tests for design of s tat ion grounds in nonuniform soil. I E E E Trans., PAS 69 (1963) 966--970

4. Jakobs, A . I . ; Kostruba, C . I . ; ~ivago, V. T.: Ras~et slo~nyh zazemljajug~ih ustrojs tv s p o m o ~ u ECVM Elek- tri~estvo Nr. 8, S. 2l --28 (1967) Moscow

5- Stag, O. ; E1-Abiad, A. : Computer methods in power system analysis. New York, London: Me Graw-Hill t968

6. Burgsdorf, V .V. : Ras6et zazemlenii v neodnorodnyh gruntah., Elektri~estvo Nr. 1, S. t5--25, (1954) Moscow

7. Ollendorf, F. : Erdstr6me. Berlin: Springer 1928

E i n g e g a n g e n am, 21. D e z e m b e r 1971

Dr.-Ing. J. M. Nahman

Depar tment of Electrical Engineering Universi ty of Belgrade, Po Box g16 1100I Belgrade, Yugoslavia

digital calculation of earthing systems in nonuniform soil

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